By Proposition REF  and Theorem REF , it follows that there is a \(C^0\) -conjugacy \(H : \tilde{X} \rightarrow G / \Gamma \)  between the canonical lift \(\tilde{X}\)  produced in Theorem REF  and a translation action on a homogeneous space \(G / \Gamma \) . So it suffices to show that any \(C^0\)  conjugacy is \(C^{r^{\prime }}\) .
The coarse Lyapunov manifolds are parameterized by nilpotent group actions, which are subgroups of the full group \(G\)  in the homogeneous model (and hence act smoothly on the space \(G / \Gamma \) ). They are conjugated by \(H\)  to the actions \(N_\alpha \curvearrowright X\)  produced in Theorem REF , which act by \(C^{r^{\prime }}\)  diffeomorphisms on their orbits by Lemma REF  or REF .
By standard Lie theoretic arguments (see, e.g., [1]}, Section 5.1), since each element is a diffeomorphism, the group action is \(C^1\)  in both cases. In the \(C^\infty \)  setting, it also follows immediately that the group action is \(C^\infty \) , and therefore provides \(C^\infty \) -coordinates. Therefore, the conjugacy is \(C^\infty \)  restricted to each leaf of a coarse Lyapunov foliations.
In the \(C^{1,\theta }\) -case, we still have that \(H\)  is \(C^1\)  in along leaves of \(W^\alpha \) . Notice that if \(\left|\left| \cdot  \right|\right|_\alpha \)  is the Hölder metric on \(TW_\alpha \)  provided by REF  then it is invariant under the isometries in Proposition REF . Then \(H_*\left|\left| \cdot  \right|\right|_\alpha \)  is a norm which is invariant under right translation on each leaf of \(W^\alpha \)  in \(G / \Gamma \)  (notice that while right translation is not defined on all of \(G/ \Gamma \) , it is well-defined on each \(W^\alpha \) -leaf). In particular, \(H_*\left|\left| \cdot  \right|\right|_\alpha \)  is \(C^\infty \)  on each \(W^\alpha \) -leaf of \(G/\Gamma \) . Therefore, \(H : W^\alpha (x) \rightarrow W^\alpha (H(x))\)  is an isometry between a Hölder metric on the leaf in \(\tilde{X}\)  and \(C^\infty \)  metric on the leaf in \(G/ \Gamma \) , and is therefore \(C^{1,\theta }\)  by Theorem REF .
The conjugacy \(H\)  is clearly \(C^{r^{\prime }}\)  along the \(\mathbb {R}^k \times K \times M\) -orbits. The smoothness of \(H\)  will now follow from iterated applications of Theorem REF . Fix a regular element \(a\) . We show that \(H\)  is \(C^{r^{\prime }}\)  along the foliations \(W^s_a\) , \(W^u_a\)  and \(\mathbb {R}^k \times M \times K\) . Then by Theorem REF , since \(\mathbb {R}^k \times M \times K\)  and \(W^s_a\)  foliate the center-stable manifold as transverse subfoliations, it will follow that \(H\)  is \(C^{r^{\prime }}\)  along the center-stable foliation. Then again, since the conjugacy is \(C^{r^{\prime }}\)  along the center-stable foliation and the unstable foliation, another application Theorem REF  shows that \(H\)  is \(C^{r^{\prime }}\) .
So we must show that \(H\)  is \(C^{r^{\prime }}\)  along the foliations \(W^s_a\)  and \(W^u_a\) . We show it for \(W^s_a\) , the proof for \(W^u_a\)  follows by considering \(-a\) . List the coarse exponents \(\Delta ^-(a) = \left\lbrace \alpha _1,\dots ,\alpha _n \right\rbrace \)  is a circular ordering, so that there are foliations \(\mathcal {W}_i\)  such that \(T\mathcal {W}_i = \bigoplus _{j=1}^i TW^{\alpha _j}\) . These foliations are obtained as intersections of stable manifolds from different Weyl chambers. We claim that \(H\)  is \(C^{r^{\prime }}\)  along \(\mathcal {W}_i\)  by induction on \(i\) .
We have already established the base case of \(i = 1\) . Assume that \(H\)  is \(C^{r^{\prime }}\)  along \(\mathcal {W}_i\) . Then by construction, \(\mathcal {W}_{i+1}\)  is a foliation whose leaves have two transverse subfoliations: \(\mathcal {W}_i\)  and \(W^{\alpha _{i+1}}\) . Since we know that \(H\)  is \(C^{r^{\prime }}\)  along each by induction and since \(W^{\alpha _{i+1}}\)  is a coarse Lyapunov foliation, it follows that \(H\)  is \(C^{r^{\prime }}\)  along \(\mathcal {W}_{i+1}\)  (again by Theorem REF ). This proves the inductive step. Since \(W^s_a = \mathcal {W}_n\) , it follows that \(H\)  is \(C^{r^{\prime }}\)  along \(W^s_a\)  and therefore, \(H\)  is \(C^{r^{\prime }}\) .
